Under the interval-valued hesitant fuzzy information environment, we investigate a multiattribute group decision making (MAGDM) method with continuous entropy weights and improved Hamacher information aggregation operators. Firstly, we introduce the axiomatic definition of entropy for interval-valued hesitant fuzzy elements (IVHFEs) and construct a continuous entropy formula on the basis of the continuous ordered weighted averaging (COWA) operator. Then, based on the Hamacher
Fuzzy sets (FSs) [
Since IVHFS was introduced, it has been used to deal with many problems, especially the MAGDM problems, one of the ways is aggregating the DMs’ opinions under each attribute for alternatives and then obtaining the collective of attribute values for each alternative [
Entropy is one of the important research topics in the fuzzy theory, which has been widely used in practical applications [
From the above analysis, we can see that IVHFS is a very useful tool to cope with uncertainty. More and more decision making methods and theories have been developed on the basis of IVHFSs. On the one hand, just as the HFSs, introducing the axiomatic definition of entropy and investigating some entropy formulas for IVHFSs are the important issues. On the other hand, more and more multiattribute GDM methods and theories have been developed with the Hamacher
In this paper, the axiomatic definition of entropy and an entropy formula for IVHFEs are investigated, and then some new improved Hamacher aggregation operators are proposed to aggregate interval-valued hesitant fuzzy information. A MAGDM approach is developed, which is based on the entropy weights and the Hamacher information aggregation operators.
To do this, the rest of the paper is organized as follows. In Section
In this section, we furnish a brief review on some basic concepts, including IVHFSs and Hamacher
Chen et al. [
Let
Notice that the number of values in different IVHFEs may be different. Suppose that
(R1) All the elements in each IVHFE
(R2) If
(R3) For convenience, we assume that all the IVHFEs have the same length
In order to compare among the different IVHFEs, we first give the properties of interval numbers.
Let
Let
For an IVHFE
In the following, we recall the triangular norm and conorm, which is an important notion in fuzzy set theory.
A function
The corresponding
For many
In particular, when
By using Hamacher
Let
Based on the above operations, Li and Peng [
Let
Let
Let
In this section, we introduce the axiomatic definition of entropy for IVHFEs and then construct an interval-valued hesitant fuzzy continuous entropy formula on the basis of COWA operator.
The COWA operator was developed by Yager [
A COWA operator is a mapping
Denoting
In what follows, we first present the axiomatic definition of entropy for IVHFEs and then investigate a continuous entropy formula for IVHFEs.
An entropy on IVHFE
or
Note that if the IVHFE
The above conditions are equivalent to the requirements of the axiomatic definition of entropy for hesitant fuzzy elements proposed by Xu and Xia [
Based on the COWA operator, we construct an information measure formula for IVHFE
In what follows, we prove that
The mapping
In order for (
Let
If
(E1) If
On the other hand, assume that
From the above analysis, we know that (
(i) If
Let
(ii) If
(E2) Suppose that
On the other hand, from the above analysis, we have
(E3) As
Since
(E4) Assume that
Similarly, if
Suppose that
In this section, we first point out that the IVHFHOWA operator and IVHFHOWG operator proposed by Li and Peng [
Suppose that
On the other hand, by the IVHFHOWG operator in Definition
Example
Let
According to Definition
Let
If
If
In the following, motivated by the geometric mean [
Let
If
If
In what follows, we discuss some desirable properties of the I-IVHFHOWA operator and I-IVHFHOWG operator.
Let
Since
Similarly, we have
It can be easily proved that the I-IVHFHOWA and I-IVHFHOWG operators have the following property.
Let
In what follows, we analyze the relationship among the I-IVHFOWA operator, I-IVHFOWG operator, I-IVHFEOWA operator, and I-IVHFEOWG operator.
In order to study the relationship among these interval-valued hesitant fuzzy information aggregation operators, we first introduce the following lemmas.
Let
Let
Let
Since
Similarly, we also get that
Then according to Definitions
Let
Since
Similarly, we have
According to Definitions
Let
Since
Note that
Using Lemma
Similarly, we have
According to Definitions
According to Theorems
Let
Let
In the following, we propose two approaches to determine the weight vector of attributes based on the continuous entropy.
Considering the entropy of the attribute
According to the entropy theory, the entropy of an attribute is smaller across alternatives, which implies that it can provide decision makers with the effective information, and the attribute should be assigned a bigger weight.
In the process of decision making, sometimes, the information about attribute weights is completely unknown or incompletely known because of lack of knowledge or data, the influence of the decision environment, and the expert’s limited expertise about the problem domain.
If the information about weight
If the information about weight
From the above analysis, we investigate a MAGDM method under interval-valued hesitant fuzzy environment; the following steps are involved.
If all the attributes
Utilize (
Based on the interval-valued hesitant fuzzy decision matrix
Calculate the score functions
Rank the overall IVHFEs
Select the priority of the alternatives according to the ranking of
End.
Emergency risk management (ERM) is a process which involves dealing with risks to the community arising from emergency events. Emergency management evaluation as one of the important parts of ERM aims at evaluating and improving social preparedness and organizational ability of an emergency operating center (EOC) in identifying, analyzing, and treating emergency risks to the community arising from emergency events [
Suppose that there are four EOCs There are five attributes to evaluate four EOCs, including The attribute weight vector The fuzzy decision matrix
Interval-valued hesitant fuzzy decision matrix.
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The proposed MAGDM method is used to rank the EOCs and get the most desirable EOC. The following steps are involved.
Considering that all the attributes
Without loss of generality, take the BUM function
Let
The overall IVHFEs of the candidates by utilizing different operators.
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I-IVHFHOWA | I-IVHFHOWG | I-IVHFHOWA | I-IVHFHOWG | |
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Calculate the score functions
The score functions obtained by the I-IVHFHOWA operator and I-IVHFHOWG operator.
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I-IVHFHOWA |
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I-IVHFHOWG |
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Rank the overall IVHFEs
Ordering of the overall IVHFEs.
Ordering | ||
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I-IVHFHOWA |
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I-IVHFHOWG |
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I-IVHFHOWA |
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I-IVHFHOWG |
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Rank the EOCs.
According to the ordering of the overall IVHFEs
Ordering of the EOCs.
Ordering | ||
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I-IVHFHOWA |
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I-IVHFHOWA |
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I-IVHFHOWA |
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I-IVHFHOWA |
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Moreover, we can analyze how the different parameter value
Lower limits of scores for EOCs obtained by I-IVHFHOWA operator.
Upper limits of scores for EOCs obtained by I-IVHFHOWA operator.
Lower limits of scores for EOCs obtained by I-IVHFHOWG operator.
Upper limits of scores for EOCs obtained by I-IVHFHOWG operator.
Deviation values of lower limits of scores for EOCs obtained by I-IVHFHOWA operator and I-IVHFHOWG operator.
Deviation values of upper limits of scores for EOCs obtained by I-IVHFHOWA operator and I-IVHFHOWG operator.
As can be seen, depending on the different parameter values
In what follows, we apply the IVHFHOWG operator (
By using the proposed interval-valued hesitant fuzzy entropy formula, Hu and Zhou [
Through the above example, we find that compared with the methods developed by Li and Peng [
In FSs theory and modern decision making science in the uncertain environment, interval-valued hesitant fuzzy MAGDM is an important research field. In this paper, we first presented the axiomatic definition of entropy for IVHFEs and constructed a continuous entropy formula for IVHFE to determine the entropy weights with respect to the decision matrix. Then, based on the Hamacher
However, there are still lots of work to be done in the future, including how to investigate sensitivity analysis and to give some practical applications of our method to the fields of decision making and supply chain management and medical diagnosis.
The authors declare that they have no conflicts of interest.
This work was supported by National Natural Science Foundation of China (no. 51204026) and Ministry of Public Security Key Laboratory of Building Fire Engineering Technology (no. KFKT2015ZD03).